No. 996

Title:

Joint distributions of waiting time random variables for patterns

Author(s):

Aki, Sigeo (Kansai University);
Hirano, Katuomi (The Institute of Statistical Mathematics)

Key words:

Discrete pattern, waiting time, generalized probability generating function, conditional expectation, discrete distribution theory.

Abstract:

    Exact joint distributions of waiting times for two patterns in a sequence of $\ell $-th order time-homogeneous Markov dependent multi-state trials are studied, where the patterns are not necessarily assumed to be distinct with each other. We prove that exact joint probability generating functions, which are regarded as expectations of the corresponding random variables, are derived through calculating the conditional expectation based on conditioning by the sooner waiting time and the pattern which comes sooner. We also give illustrative numerical examples in order to demonstrate the performance of our results.